Question: Find the number of positive integers $n,$ $1 \le n \le 1000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients.
Solution: If $x^2 + x - n$ factors as the product of two linear factors with integer coefficients, then it must be of the form
\[(x - a)(x - b) = x^2 - (a + b)x + ab,\]where $a$ and $b$ are integers.  Then $a + b = -1$ and $ab = -n$, which means $n = -ab = -a(-a - 1) = a(a + 1).$  We want $1 \le n \le 1000.$  The possible values of $a$ are then 1, 2, $\dots,$ 31, so there are $\boxed{31}$ possible values of $n.$  (Note that $a$ can also be $-32,$ $-31,$ $\dots,$ $-2,$ but these give the same values of $n.$)